I think Amir was trying to tailor the video to audio consumers, and in an effort to dumb it down, the way current and voltage are described is not 100% accurate. I used to design exhibits for science museums, and did quite a few solid state high voltage static electricity exhibits in the range of 5-200kV. We had to consult experts and submit research papers to the museums along with our exhibits.
Yes, as a layman explanation, voltage is the one to watch out for, since generally we don't know if it is current limited. And current is drawn as necessary, so high current capacity at low voltage is not something to worry about.
But the real answer is that V / R(system) + R(body) = I (through our body). Some nuance is needed to determine if we are talking about current through our heart across 2 arms, or between voltage source and earth. But generally take the voltage, divide by your body's resistance, and make sure it is not more than 2-30mA max. You can actually hold 2 probes of a multimeter to get a reasonable resistance measurement. Try holding the multimeter probe with your hands wet vs dry. People use 2000 Ohms as as nominal resistance in the body, but it can actually be 10 - 100X this depending on humidity and other factors, including how big or small you are, what you're wearing, etc. Take 48V and divide by 2000 ohms and you get 24mA. That's why 48V is kinda sorta still low enough voltage to not be life threatening, but you still gotta be careful. Also don't do any work with your car battery if you are standing in a pool of water.
Generally, a safe guideline is that for physical contact to an electrical source of 2 seconds or less 2mA - 30mA is safe regarded as safe. For more than 10 seconds, which is regarded as continuous, 20-30mA is regarded as safe, but not advisable. There is actually a bit of variation here in terms of what may or not kill or hurt someone. But 2-30mA is generally the range that is discussed.
As a starting point the International Electrotechnical Commission (IEC) standards for Domestic Electrical Appliances 60335 "Safety for Domestic Electrical and Similar Electrical Appliances Safety General Requirements." Using clause 8.1.4, defines safe discharge of electricity in 2 levels: 1) @voltages from 45V to 450V shocks should be limited to the energy stored in capacitance of 0.1uF into 2000Ohms 2) At voltages up to 15kV, the delivered charge should not exceed 45uC into 2000 ohms (nominal human impedance). Ref is from IEC60335-1:2002. In addition, this reference goes on to say that for DC currents, peak should not exceed 2mA. These limits are regarded as "Safe Limits," for inadvertent or accidental contact and not necessarily for deliberate or prolonged contact.
Other references:
See IEC 60335, IEC 60664. In particular, see IEC 60990 column 1999 "Methods of Measurement of Touch Current and Protective Conductor Current." This is a very good paper, and it discusses a human body model. While measuring current through the body, this model is very good because it takes into account how different frequencies affect the body. Also see IEC 60479-1, part 2 "Special Aspects Involving Human Beings."
Bottom line though, it is current is what kills, but it's a long explanation so we usually shorten it and say watch out for high voltage.
Lastly, I'd like to give an imperfect analogy. Imagine you want to throw a brick off a cliff. The height is the voltage, and the fluid in which it is dropped is the resistance. If you drop a brick off a cliff in the air then that brick is going to hurt and kill someone. Now imagine that instead of air, the cliff is underwater and water is the fluid. That brick would not hurt someone. That's because there was sufficient resistance in the system, which rendered the voltage harmless.